# Commuting metric past Dirac spinors?

• auditor
In summary, the conversation discusses how to go from the equation i\mathcal{M} = {\overline{v}^s^'} (p^{'}) (-ie\gamma^\mu)u^s(p) \left( \frac{-ig_{\mu\nu}}{q^2} \right) \overline{u}^r (k) (-ie\gamma^\nu) v^{r^{'}} (k) at the bottom of page 131 to (5.1) at the top of 132, which is i\mathcal{M} = \frac{ie^2}{q^2}(\overline{v}(p^{'}) \gamma^\
auditor
I'm wondering how in Peskin & Schroeder they go from

$$i\mathcal{M} = {\overline{v}^s^'} (p^{'}) (-ie\gamma^\mu)u^s(p) \left( \frac{-ig_{\mu\nu}}{q^2} \right) \overline{u}^r (k) (-ie\gamma^\nu) v^{r^{'}} (k)$$

at the bottom of page 131 to (5.1) at the top of 132 which reads

$$i\mathcal{M} = \frac{ie^2}{q^2}(\overline{v}(p^{'}) \gamma^\mu u(p) (\overline{u}(k)\gamma_\mu v(k^{'}))$$

Most of the stuff is ok, in particular dropping the spin superscripts. But how does the metric commute with

$$\overline{u}^r (k)$$

? I kind'a remember that the spinors are elements of the SU(2) group and that this might be related to my question. It seems as though the commutator is 0. But if I write out the metric and the Dirac spinor on matrix and vector form respectively I get a matrix product of the form

$$(4 x 4)\cdot (1 x 4)$$

which is undefined. I suspect by doing this I'm mixing apples and oranges, thus my reference to the SU(2) structure. I really don't have time to dwell in group theoretical details right now, although I'm aware that this is the only way to really get QFT. Could anyone please advice? My intuition is that, yes; they do commute, but I want to be sure.

Thanks!

You're thinking too hard.

$g_{\mu\nu}$ is not the metric, it is (for each $\mu$ and $\nu$) a component of the metric, i.e., a number, and, as such, commutes with everything!

Wow - that's true. :) Thanks a lot George, really appreciate it!

## 1. What is a commuting metric?

A commuting metric is a mathematical concept used in physics to describe the properties of a space. It is a metric tensor that satisfies the condition that its components commute with each other, meaning that the order in which they are multiplied does not affect the final result.

## 2. What are Dirac spinors?

Dirac spinors are mathematical objects used in quantum field theory to describe the behavior of particles with spin. They are four-component complex vectors that represent the wave functions of particles, and they satisfy the Dirac equation which describes the dynamics of fermions.

## 3. How are commuting metric and Dirac spinors related?

The commuting metric is used to define the inner product of Dirac spinors in a space. This inner product is necessary for calculating physical quantities such as probabilities and amplitudes in quantum field theory. The Dirac equation also involves the use of the commuting metric in its formulation.

## 4. What is "commuting metric past" in the context of Dirac spinors?

"Commuting metric past" refers to the use of a commuting metric in the past light cone of a point in spacetime. This concept is important in the study of causality and the behavior of particles in space and time.

## 5. Why is the study of commuting metric past Dirac spinors important?

The study of commuting metric past Dirac spinors is important in understanding the behavior of particles in quantum field theory, as well as in studying the effects of causality and the structure of spacetime. It is also relevant in the development of theories such as general relativity and the standard model of particle physics.

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